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Perfect state transfer on graphs with clusters

Hermie Monterde, Hiranmoy Pal

TL;DR

This work develops a cluster-based framework for perfect and pretty good quantum state transfer on graphs under the adjacency, Laplacian, and signless Laplacian Hamiltonians. By introducing the graph-with-clusters construction $G(H)$ and analyzing how PST/PGST lift from a base graph $H$ to $G(H)$, the authors generate infinite families of graphs (including non-regular ones) that admit pair state transfer between the same pair of states at the same time, across all three Hamiltonians. They provide concrete results for complete graphs, their edge-removals, and complements, and connect the transfer properties to coherent algebras, sequential joins, and a rich set of graph products (Cartesian, corona, blow-ups, and lexicographic). The framework yields broad, transferable criteria for constructing and understanding PST/PGST in complex networks, with implications for robust quantum information routing on non-regular graphs.

Abstract

Using graphs with clusters, we provide a unified approach for constructing graphs with pair state transfer-relative to the adjacency, Laplacian, and signless Laplacian matrix-between the same pair of states at the same time, despite being non-regular. We show that for each $k\geq 5$, there are infinitely many connected graphs with maximum valency $k$ admitting this property. This framework also aids in establishing sufficient conditions for pair state transfer in edge-perturbed graphs, including complete graphs and complete bipartite graphs. Furthermore, we utilize graph products to generate new infinite families of graphs with the above property.

Perfect state transfer on graphs with clusters

TL;DR

This work develops a cluster-based framework for perfect and pretty good quantum state transfer on graphs under the adjacency, Laplacian, and signless Laplacian Hamiltonians. By introducing the graph-with-clusters construction and analyzing how PST/PGST lift from a base graph to , the authors generate infinite families of graphs (including non-regular ones) that admit pair state transfer between the same pair of states at the same time, across all three Hamiltonians. They provide concrete results for complete graphs, their edge-removals, and complements, and connect the transfer properties to coherent algebras, sequential joins, and a rich set of graph products (Cartesian, corona, blow-ups, and lexicographic). The framework yields broad, transferable criteria for constructing and understanding PST/PGST in complex networks, with implications for robust quantum information routing on non-regular graphs.

Abstract

Using graphs with clusters, we provide a unified approach for constructing graphs with pair state transfer-relative to the adjacency, Laplacian, and signless Laplacian matrix-between the same pair of states at the same time, despite being non-regular. We show that for each , there are infinitely many connected graphs with maximum valency admitting this property. This framework also aids in establishing sufficient conditions for pair state transfer in edge-perturbed graphs, including complete graphs and complete bipartite graphs. Furthermore, we utilize graph products to generate new infinite families of graphs with the above property.
Paper Structure (9 sections, 25 theorems, 13 equations, 6 figures)

This paper contains 9 sections, 25 theorems, 13 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Graphs with clusters
  3. PST between pure states
  4. Pair state transfer
  5. Complete graph
  6. Coherent algebra
  7. Sequential join
  8. Complement
  9. More graph products

Key Result

Lemma 1

Let ${\bf x}\in{\mathbbm R}^c$ satisfying ${\mathbf 1}_c^T{\bf x}=0,$ and $H$ be a graph for which ${\mathbf 1}_c$ is an eigenvector of $M(H)$. If $G(H)$ is the graph in df2, then $\delta {\mathbf 1}_s^T{\bf z}+\sigma_{{\bf x}}(H)=\sigma_{\widetilde{{\bf x}}}(G(H)).$

Figures (6)

  • Figure 1: The graphs $G_1$ (left) and $G_2$ (right)
  • Figure 2: A graph exhibiting Laplacian pair state transfer.
  • Figure 3: The graph $P_2 \square (K_1 \vee C_4)$
  • Figure 4: Vertex corona
  • Figure 5: Edge corona
  • ...and 1 more figures

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Theorem 2
  • Theorem 3
  • ...and 22 more